Why does Microsoft not allow you to link a build configuration to a publish profile. Instead it tells you to use the dropdown in the main VS interface.
To get around this, you can write a batch script to do both parts of the process. Use MSBuild to build your projects in the required build configuration (we use release for our web apps). If all you need to do is copy the output, MSBuild should be able to do that for you as well with a copy task. If you need to deploy a web application, use MSDeploy command line tools to complete the publish command. Going command line with both of these tools opens up a lot of customization.